identification of space-time distributed parameters …trenchea/papers/identificationrd.pdf ·...

IDENTIFICATION OF SPACE-TIME DISTRIBUTED PARAMETERSIN THE GIERER-MEINHARDT REACTION-DIFFUSION SYSTEM

MARCUS R. GARVIE∗ AND CATALIN TRENCHEA†

Abstract. We consider parameter identification for the classic Gierer-Meinhardt reaction-diffusion system. The original Gierer-Meinhardt model [A. Gierer and H. Meinhardt, Kybernetik,12 (1972), pp. 30-39] was formulated with constant parameters and has been used as a prototypesystem for investigating pattern formation in developmental biology. In our paper the parametersare extended in time and space and used as distributed control variables. The methodology employsPDE-constrained optimization in the context of image-driven spatiotemporal pattern formation. Weprove the existence of optimal solutions, derive an optimality system, and determine optimal solu-tions. The results of numerical experiments in 2D are presented using the finite element method,which illustrates the convergence of a variable-step gradient algorithm for finding the optimal param-eters of the system. A practical target function was constructed for the optimal control algorithmcorresponding to the actual image of a marine angelfish.

Key words. optimal control theory, parameter identification, reaction-diffusion equations,image-driven optimization, variable-step gradient algorithm, finite element method, Gierer-Meinhardt,pattern formation

AMS subject classifications. 49J20, 93B30, 35K57, 35B36

1. Introduction. Many processes in the applied sciences are adequately mod-eled by partial differential equations [19]. However, it is not always possible to measureor calculate the model parameters with the necessary accuracy, particularly in livingorganisms. In these cases mathematical techniques for estimating model parametersbecome important. Parameter estimation is also an important technique for identify-ing redundant parameters (and hence the key parameters) in complex systems withmany parameters. The need to identify parameters in evolution equations arises inmany disciplines, including biology, physics, engineering, and chemistry. Much re-search has been devoted to the development of computational methods for estimatingparameters in such equations (see, for example [1, 2, 3, 4, 5, 10, 11, 14, 18, 31, 36, 46,47, 54, 64] and the references therein). The typical procedure involves integrating theevolution equation to obtain a simulation result that is compared to an observed dataset, and then the application of least squares techniques to minimize a cost functionalwith respect to parameters in an admissible set [6, 21, 30, 34, 38, 40, 41, 51].

There are relatively few works that focus on parameter identification for reaction-diffusion (RD) equations [23, 25, 27, 28, 35], which is a fertile and growing area ofresearch with many applications, for example, population dynamics [16, 39], synaptictransmission at a neuromuscular junction [13], color negative film development [20],chemotaxis [17], epidemiology [37], and brain tumor growth [32].

Traditional studies of RD equations have focussed on models with homogeneousparameters, i.e. parameters that are constant in time and space. In reality, parametersoften operate in heterogeneous environments. A number of authors have consideredRD models with spatially varying parameters [8, 42, 48, 49], or temporally varyingparameters [57, 62, 65], but to our knowledge none allow the parameters to vary freelyin time and space. By relaxing this assumption, and allowing the parameters to vary

∗Corresponding author. Department of Mathematics and Statistics, MacNaughton Building, Uni-versity of Guelph, Guelph, ON Canada N1G 2W1 ([email protected]).†Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, PA 15260

([email protected]).

1

2 MARCUS R. GARVIE AND CATALIN TRENCHEA

in time and space, we apply an image-driven methodology for parameter identificationin RD equations with broad applicability. Another novel aspect of our work is thatthe image used as data for the optimal control procedure is taken directly from nature.

For concreteness, we illustrate our method by considering the identification prob-lem for the two-component activator-inhibitor system of RD equations introduced byGierer and Meinhardt [26] for pattern formation. The Gierer-Meinhardt system is oneof the most famous models in biological pattern formation. The original formulationhas fixed parameter values. We consider the more general situation where two keyparameters, µ and α, depend on space x and time t. In non-dimensional form [48]the RD system has the following form

∂u∂t = Du∆u+ ru2

v − µ(x, t)u+ r,∂v∂t = Dv∆v + ru2 − α(x, t)v,

(1.1)

with non-negative parameters r,Du, and Dv, and non-negative morphogen concen-trations u(x, t) and v(x, t). ∆ denotes the standard Laplacian operator in two spacedimensions.

(a) (b)

Fig. 1.1: Numerical solutions u of (1.1) at time T = 500 with Dv = 0.27, Du = 9.45 × 10−4,r = 0.001, α = 100, with (a) µ = 2.5, and (b) µ = 2.5 within a circle of radius 0.9, centre (1, 1), andµ = 1.5 elsewhere. Spatial discretization step = 0.01. An IMEX Galerkin finite element method withpiecewise linear continuous basis functions was employed with first and second order time-steppingschemes 1-SBDF and 2-SBDF [56], with time steps 1 × 10−8 and 0.01 respectively. HomogeneousNeumann boundary conditions were employed. Initial data u0 and v0: small random perturbations(±10%) of the steady state solutions.

When the parameters in the Gierer-Meinhardt model are appropriately chosen,the mechanism of ‘diffusion induced instability’, or Turing mechanism [63], leads tomorphogen concentrations with characteristic regular spacing of peaks and troughs(a pattern) [26]. The identification problem for the case of constant parameters µ, αwas treated in [23]. When the parameters µ and α are allowed to vary in space therange and complexity of possible patterns increases [49]. Typical numerical solutionsof the Gierer-Meinhardt system (1.1) for spatially homogeneous and inhomogeneousparameters are shown in Figure 1.1.

The remainder of this paper is structured as follows. In Section 2 the well-posedness of the direct problem is discussed, while in Section 3 a cost functional isdefined that allows us to setup the inverse problem. In Section 4 we establish the

PARAMETER IDENTIFICATION IN REACTION-DIFFUSION SYSTEMS 3

existence of optimal solutions, derive an optimality system, and determine optimalsolutions. In Section 5 we give second-order optimality conditions. In Section 6 thenumerical methods are discussed, including the statement of the discrete optimalitysystem and the construction of a discrete target function. In Section 7 the results ofnumerical experiments are presented that illustrate the convergence of a variable-stepgradient algorithm for finding optimal parameters. Finally, in Section 8 we make someconclusions.

2. Well-posedness of the direct problem. Before stating the well-posednessresult for the Gierer-Meinhardt model we need to establish the formal setting andrestate the RD system with appropriate initial and boundary conditions. Let Ω be abounded and open subset of Rd, d ≤ 2, with a Lipschitz continuous boundary, andlet Q := Ω × (0, T ) be the space-time cylinder. The direct problem is formulated asfollows:

Find the morphogen concentrations u(x, t) and v(x, t) such that

∂u

∂t= Du∆u+ ru2

v− µ(x, t)u+ r in Q, (2.1a)

∂v

∂t= Dv∆v + ru2 − α(x, t)v in Q, (2.1b)

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω (2.1c)∂u

∂ν= ∂v

∂ν= 0 on ∂Ω× (0, T ), (2.1d)

where the fixed parameters r,Du, and Dv are positive, and u(x, t) and v(x, t) arethe morphogen concentrations defined for (x, t) ∈ Q. Du and Dv are the diffusioncoefficients of u and v respectively, ∆ =

∑di=1 ∂

2/∂x2i denotes the standard Laplacian

operator in d ≤ 2 space dimensions, and ν denotes the outward normal to ∂Ω, theboundary of Ω. We assume that µ(x, t) and α(x, t) are bounded, Lipschitz continuousfunctions on Q, which we denote by µ, α ∈ Lip(Q), belonging to the set of admissibleparameters:

Uad :=

(µ, α) ∈ Lip(Q)2, 0 ≤ µ(x, t) ≤ µ1, 0 ≤ α(x, t) ≤ α1, ∀(x, t) ∈ Q,

(2.2)for some finite real numbers µ1, α1. We assume zero flux of the morphogen concen-trations across the boundary and that the initial concentrations are continuous andpositive, with

u0(x) ≥ umin0 > 0, v0(x) ≥ vmin0 > 0 for all x ∈ Ω. (2.3)

Theorem 2.1. Let (µ, α) ∈ Uad and u0, v0 ∈ C(Ω) satisfying (2.3). Then thereexists a unique global positive classical solution of the Gierer-Meinhardt RD system(2.1a)-(2.1d).

Proof. The existence of a unique global classical solution of the system (2.1a)-(2.1d) follows from [55]. To prove the nonnegativity of solutions observe that thereaction kinetics in (2.1a)-(2.1b), denoted f and g respectively, satisfy

f(0, v), g(u, 0) > 0 for all u, v > 0,

and the initial data (u0(x), v0(x)) is in (0,∞)2 for all x ∈ Ω. Thus by a maximumprinciple [58, Corollary 14.8] the solution (u(x, t), v(x, t)) lies in [0,∞)2 for all x ∈ Ω


and for all t > 0 for which the solution of (2.1a)-(2.1d) exists. In other words [0,∞)2

is positively invariant for the system.For the positivity of the solutions satisfying (2.1a)-(2.1d) let u = u(x, t), v =

v(x, t) be solutions of the simplified system

∂u

∂t−Du∆u+ µ(x, t)u = r, (2.4a)

∂v

∂t−Dv∆v + α(x, t)v = ru2, (2.4b)

with homogeneous Neumann boundary conditions and initial conditions

u(x, 0) = u0(x), v(x, 0) = v0(x) for all x ∈ Ω.

Applying the comparison principle (see e.g. [55, 52]) to (2.4a) we deduce u > 0 andsimilarly from (2.4b) we infer that v > 0. Subtracting (2.4a)-(2.4b) from (2.1a)-(2.1b)we obtain

∂(u− u)∂t

−Du∆(u− u) + µ(x, t)(u− u) = ru2

v,

∂(v − v)∂t

−Dv∆(v − v) + α(x, t)(v − v) = r(u2 − u2).

Using the comparison theorem once again and the nonnegativity of v in the firstequation above we obtain that u > u on Ω × (0, T ). Then from the second equationusing the same theorem we conclude that the solution to the Gierer-Meinhardt system(2.1a)-(2.1d) has a positive lower bound

u(x, t) > u(x, t) > 0, v(x, t) > v(x, t) > 0 in Ω× (0, T ). (2.5)

Lemma 2.2. The solution (u, v) of system (2.1a)-(2.1d) can be estimated, withrespect to µ, α, uniformly from below by

u(x, t) ≥ U(t) := r

µ1(1− e−µ1t) + umin0 e−µ1t,

v(x, t) ≥ V (t) := r

∫ t

0eα1(s−t)U2(s)ds+ vmin0 e−α1t,

in Ω× [0, T ].Proof. After lengthy, but elementary calculation, it can be shown that U(t) and

V (t) satisfy the following system of ordinary differential equations

U ′ + µ1U = r,

V ′ + α1V = rU2,

with the initial condition U(0) = umin0 , V (0) = vmin0 . Because u, U satisfy

∂(u− U)∂t

−Du∆(u− U) + µ(x, t)(u− U) = (µ1 − µ(x, t))U ≥ 0,

and U(t) > 0, by a maximum principle [52] we deduce that u ≥ U in Ω× (0, T ).Since V (t) > 0, a similar argument holds for v and V , and we have

∂(v − V )∂t

−Dv∆(v − V ) + α(x, t)(v − V ) = (α1 − α(x, t))V + r(u2 − U2) ≥ 0.


Finally, from (2.5) we obtain the lower bounds on u and v.Remark 2.1. From the proof of Lemma 2.2 we deduce that v(x, t) ≥ vmin0 e−α1T

on Q.

3. Setup of the inverse problem. For the inverse problem we are given pos-sibly perturbed measurements (u, v) corresponding to the state variables (u, v) andseek parameters µ, α such that (u, v) best approximates (u, v).

For given T > 0, u0, v0 ∈ C(Ω), and u, v ∈ L2(Q) not necessarily a solutionof (2.1a)-(2.1d), the least-squares approach leads to the minimization of the costfunctional

J (µ, α) = 12

∫ T

0

∫Ω

(β1|uµ,α − u|2 + β2|vµ,α − v|2

)dx dt

+12

∫Ω

(γ1|uµ,α(x, T )− u(x, T )|2 + γ2|vµ,α(x, T )− v(x, T )|2

)dx,

where (uµ,α, vµ,α) is the solution of (2.1a)-(2.1d) that corresponds to (µ, α). Inverseproblems related to partial differential equations are usually ill-posed (see e.g. [61, 33])and thus the least-squares approach is not numerically sufficient. To circumvent thisproblem we use Tikhonov regularization [15], which yields the following minimizationproblem:

(P) Find (µ∗, α∗) ∈ Uad such that

J (µ∗, α∗) = infµ,α∈Uad

J (µ, α),

(P) where

J (µ, α) = J (µ, α) + 12

∫ T

0

∫Ω

(δ1µ

2 + δ2α2) dx dt. (3.1)

The terms weighted by βi measure the discrepancy between the solution andmeasurements over the space-time cylinder Q, while the weights γi assign varyingemphasis on the match at the final time T . The terms weighted by δi effectivelybound the size of the key parameters µ and α and allow for possibly noisy data.

4. The minimization problem.

4.1. Existence of an optimal solution. We establish the existence of an op-timal solution of the minimization problem (P). As before we assume Ω is an openbounded domain with Lipschitz continuous boundary ∂Ω. We denote u∗ = uµ∗,α∗

and v∗ = vµ∗,α∗ .Proposition 4.1. Given u0, v0 ∈ C(Ω), and u, v ∈ L2(Q), there exists a solution

µ∗, α∗ ∈ Uad of the minimization problem (P) s.t. (u∗, v∗) ∈ C([0, T ];L2(Ω)2) ∩L2(0, T ;H1(Ω)2).

Proof. The argument is standard [6], so we only give a sketch proof. By energy-type estimates on minimizing sequences to (P), using the uniform lower bound fromLemma 2.2, the set

(µ, α, u, v) ∈ Uad × L2(Q)2 : (2.1a)-(2.1d) holds

is closed in C(Q)2×C([0, T ];L2(Ω)2)∩L2(0, T ;H1(Ω)2). The weak lower semicontinu-ity of the function ((µ, α), (u, v)) 7→ J (µ, α) on L2(Q)2×L2(Q)2 implies the existenceof a global minimizer.


4.2. First-order necessary conditions. We derive the first-order necessaryconditions associated with the optimal control problem (P). If the Gateaux derivativeof the functional exists, then the optimal solution must satisfy this standard first-ordernecessary condition [60, 53].

Lemma 4.2. Let u0, v0 ∈ C(Ω). If (µ∗, α∗) is an optimal solution and thefunctional J (µ∗, α∗) is Gateaux differentiable, then a necessary condition for (µ∗, α∗)to be a minimizer of J (µ∗, α∗) is

dJ (µ∗, α∗)d(µ, α) · (µ− µ∗, α− α∗) ≥ 0 ∀(µ, α) ∈ Uad.

It is clear that for (µ, α) ∈ Uad the solution (u, v) is classical and hence weak. Itis convenient to work in the topology of L2(0, T ;H1(Ω)), which facilitates the useof well-known results. We recall that the solution of (2.1a)-(2.1b) defines a mappingu = uµ,α, v = vµ,α from Uad to L2(0, T ;H1(Ω)) which is Gateaux differentiable, andthus Lemma 4.2 can be applied.

Lemma 4.3. Let u0, v0 ∈ C(Ω). The map (µ, α) 7→ (uµ,α, vµ,α) from Uad toL2(0, T ;H1(Ω)2), defined as the solution of (2.1a)-(2.1b), has a Gateaux derivativein every direction (µ, α) in the tangential cone TanUad(µ, α) of Uad at (µ, α). Fur-thermore, (u, v) = ( du

d(µ,α) ,dv

d(µ,α) ) · (µ, α) is the solution of the problem

ut −Du∆u = r2uuv−u2v

v2 − µu− µu,

vt −Dv∆v = 2ruu− αv − αv, (4.1)u(0) = 0, v(0) = 0.

Proof. The argument is standard [6, 29].The Gateaux derivative provides useful information about the sensitivity of the

state system at a point (µ, α) in a particular direction (µ, α) ∈ TanUad(µ, α), butcomplete information requires one to solve (4.1) for every possible direction (µ, α).However, to minimize the functional we need only an integral over all these directionswhich can be obtained from the solution of an adjoint system.

Lemma 4.4. Let u0, v0 ∈ C(Ω), (µ, α) ∈ TanUad(µ, α) and (u, v) be defined by(4.1). For every (F,G) ∈ L2(0, T ;L2(Ω)2) we have∫ T

0

∫Ω

(Fu+Gv

)dx dt = −

∫Ω

(pu+ qv)∣∣∣T0dx−

∫ T

0

∫Ω

(µup + αvq) dx dt, (4.2)

where (p, q) is the solution of the adjoint problem

−pt −Du∆p−(

2ruv− µ

)p− 2ruq = F,

−qt −Dv∆q + ru2

v2 p + αq = G,

∂p∂ν

(x, t) = ∂q∂ν

(x, t) = 0, x ∈ ∂Ω,

p(x, T ) = pT (x), q(x, T ) = qT (x), x ∈ Ω.

(4.3)


Proof. The left-hand side of (4.2) can be evaluated by (4.3) and (4.1) via integra-tion by parts, which is justified by the regularity properties of the quantities involved.

Next we show that the optimal coefficients µ∗, α∗ in Lemma 4.2 are characterizedby the solution of a particular adjoint system.

Theorem 4.5. Let (µ∗, α∗) be an optimal solution to problem (P), u, v ∈ C(Ω),and let (p, q) be the solution of the particular adjoint problem

−pt −Du∆p−(

2ru∗

v∗− µ

)p− 2ru∗q = β1(u∗− u),

−qt −Dv∆q + ru∗2

v∗2p+ αq = β2(v∗ − v),

∂p

∂ν(x, t) = 0, ∂p

∂ν(x, t) = 0, x ∈ ∂Ω,

p(x, T ) = γ1(u∗(x, T )− u(x, T )), q(x, T ) = γ2(v∗(x, T )− v(x, T )), x ∈ Ω.(4.4)

Then we have

µ∗ = max

0,min

1δ1u∗p, µ1

, α∗ = max

0,min

1δ2v∗q, α1

. (4.5)

Proof. We compute the Gateaux derivative of the cost functional J (µ∗, α∗) inthe direction of (µ, α) ∈ TanUad(µ∗, α∗). We have

dJ (µ∗, α∗)d (µ, α) · (µ, α)

=∫Q

β1(u∗ − u)(du∗µ∗,α∗

d(µ, α) · (µ, α))

+ β2(v∗ − v)(dv∗µ∗,α∗

d(µ, α) · (µ, α))dx dt

+γ1

∫Ω

(u∗(T, x)− u(T, x))(du∗µ∗,α∗

d(µ, α) · (µ, α))

(T, x)dx

+γ2

∫Ω

(v∗(T, x)− v(T, x))(dv∗µ∗,α∗

d(µ, α) (µ, α))

(T, x)x +∫Q

(δ1µ∗µ+ δ2α∗α) dx dt

=∫Q

(−µu∗p− αv∗q + δ1µ∗µ+ δ2α

∗α) dx dt,

where (du∗µ∗,α∗/d(µ, α), dv∗µ∗,α∗/d(µ, α)) is the solution of the sensitivity system (4.1).Now from the definition of optimality in problem (P), as (µ∗, α∗) is an optimal solutionand the Gateaux derivative of the functional exists, then from Lemma 4.4∫

Q

(δ1µ∗ − u∗p)(µ− µ∗) + (δ2α∗ − v∗q)(α− α∗)dx dt ≥ 0 ∀(µ, α) ∈ Uad,

which completes the proof.

5. Second-order optimality conditions. To ensure that a solution (µ∗, α∗)satisfying the first-order optimality conditions (4.4)-(4.5) solves (P), we state withoutproof the following second-order sufficient optimality conditions (see e.g. [43, 11]).We require the following differentiability results:


Lemma 5.1. The map (µ, α) 7→ (uµ,α, vµ,α) defined in Lemma 4.3 is of class C2.Moreover, for every (µ1, α1), (µ2, α2) ∈ TanUad(µ, α),(u12, v12) = ( d2u

d(µ,α)2 ,d2v

d(µ,α)2 )(µ1, α1)(µ2, α2) is the solution of

u12t−Du∆u12 =r

(2(u1u2+uu12)v−uu1v2

v2 − (2uu2v1+u2v12)v2−2u2v1vv2

v4

)µ1u2 − µ2u1 − µu12,

v12t −Dv∆v12 = 2r (u1u2 + uu12)− α1v2 − α2v1 − αv12,(5.1)

where (ui, vi) = ( dud(µ,α) ,

dvd(µ,α) ) · (µi, αi), i = 1, 2, is the solution of problem (4.1).

The cost functional (3.1) J : Uad → R is of class C2 and

d2J (µ, α)d(µ, α)2 (µ1, α1)(µ2, α2)

=∫Q

(β1u1u2 + β2v1v2) dx dt+∫

Ω(γ1u1(T )u2(T ) + γ2v1(T )v2(T )) dx

+∫Q

p

(ru2

v2 v12 − 2ruvu12 − µ1u2 − µ2u1 + 2r (2uu2v1 + u2v12)− 2u2v1vv2

v4

)dx dt

+∫Q

q (2ru1u2 − α1v2 − α2v2) dx dt+∫Q

(δ1µ1µ2 + δ2α1α2)dx dt,

where (p, q) is the adjoint state satisfying (4.4).The sufficient second-order optimality conditions are given by the following result

(see e.g. [11, 27, 43]).Theorem 5.2. Assume that (µ∗, α∗) ∈ Uad are admissible parameters of problem

(P), (u∗, v∗) the associated states, (µ∗, α∗, u∗, v∗) satisfy (4.4)-(4.5), and there existsκ > 0 such that

d2J (µ∗, α∗)d(µ, α)2 (µ1, α1)2 ≥ κ‖(µ1, α1)‖2L2(Q) ∀(µ1, α1) ∈ TanUad(µ∗, α∗). (5.2)

Then there exist ε > 0 and δ > 0 such that for every admissible parameters (µ, α) ofproblem (P) the following inequality holds:

J (µ∗, α∗) + δ

2(‖µ∗ − µ‖2L∞(Q) + ‖α∗ − α‖2L∞(Q)) ≤ J (µ, α)

if ‖µ∗ − µ‖2L∞(Q) + ‖α∗ − α‖2L∞(Q) < ε.

If the second-order sufficient optimality condition is satisfied, then the stability ofa locally optimal solution holds and the convergence of a gradient type algorithm isguaranteed (see e.g. [12, 29]).

6. Numerical methods.

6.1. Discrete optimality system. Initially we partition the domain Ω into alarge number of approximately equilateral triangles τ using an unstructured meshgenerator. We used the automatic mesh generator MESH2D (available from http://www.mathworks.com/matlabcentral/fileexchange/), which utilizes a convenientmesh quality indicator for ensuring a good quality mesh (see (7.1) in [50]). The spatialdiscretization of the state equations and adjoint equations was undertaken using a‘lumped mass’ [59] Galerkin finite element method with piecewise linear continuous

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basis functions (e.g. [7, 9, 22, 24]). We introduce Sh, the standard Galerkin finiteelement space

Sh := χ ∈ C(Ω) : χ|τ is linear ∀τ ∈ Ω ⊂ H1(Ω).

The time discretization of RD equations for pattern formation requires careful treat-ment. This is because several popular first order accurate time-stepping schemes,as well as schemes that produce weak decay of high frequency spatial errors mayyield qualitatively misleading and incorrect results [56]. Consider a generic system ofcoupled RD equations of the following form

ut = Du∆u+ f(u, v),vt = Dv∆v + g(u, v),

where Du and Dv are the diffusion coefficients, and f and g represent the reac-tion kinetics depending on morphogen concentrations u and v. To approximate theGierer-Meinhardt system (2.1a)-(2.1b) we employed the following second order, 3-level, implicit-explicit (IMEX) scheme (2-SBDF)(in fact, in order to derive stablefinite element schemes we approximated regularized versions of these equations; see 9for further details):

3un+1−4un+un−1

2∆t −Du∆un+1 = 2f(un, vn)− f(un−1, vn−1),3vn+1−4vn+vn−1

2∆t −Dv∆vn+1 = 2g(un, vn)− g(un−1, vn−1),(6.1)

where un ≈ u(n∆t), vn ≈ v(n∆t) and ∆t is the step size. The backward differenti-ation formula 2-SBDF is recommended by Ruuth [56] as a good choice for most RDproblems. To approximate the RD system at the first time step we used a well-knownfirst order semi-implicit backward differentiation IMEX scheme (1-SBDF) [56].

The derivation of a discrete optimality system led to similar time-stepping schemesfor the linear adjoint equations (see 10). Application of the finite element method forthe spatial discretization coupled with the time-stepping schemes led to sparse linearsystems of algebraic equations, which were solved in MATLAB (R2008a) using theGMRES iterative solver.

Corresponding to the continuous cost functional (3.1) the discrete cost functionalis given by

JN = ∆t2

N∑n=1

(β1‖unh − unh‖2L2(Ω) + β2‖vnh − vnh‖2L2(Ω)

)+1

2

(γ1‖uNh − uNh ‖2L2(Ω) + γ2‖vNh − vNh ‖2L2(Ω)

)(6.2)

+∆t2

N∑n=1

(δ1‖µnh‖2L2(Ω) + δ2‖αnh‖2L2(Ω)

),

for discrete parameters µnh, αnh ∈ Lip(Ω), 0 ≤ µnh(x) ≤ µ1, 0 ≤ αnh(x) ≤ α1, discretetarget functions unh, vnh ∈ Sh, and discrete states unh, vnh ∈ Sh. Letting πh be theLagrange interpolation operator such that πh : C(Ω) 7→ Sh, the initial states weretaken as u0

h = πhu0(x), v0h = πhv0(x).

We can now formulate the fully discrete optimal control problem as:


(Ph,∆t) Given a time step of ∆t = T/N , a maximum triangle size h,u0, v0 ∈ H1(Ω) ∩ L∞(Ω) and unh, v

nh ∈ Sh, find (unh, vnh , µnh, αnh)

∈ Sh×Sh×Sh×Sh such that (6.1) is satisfied for n = 1, 2, . . . , Nand the cost functional (6.2) is minimized.

The construction of the discrete target functions unh, vnh is described below (Section6.2). Let pnh, qnh ∈ Sh denote the fully discrete approximations to the adjoint variablesp, q, then corresponding to (4.5) the following theorem holds, which characterizes therelationship between discrete adjoint variables and discrete parameter values:

Theorem 6.1. If u(n)h , v

(n)h Nn=0 and α(n)

h , µ(n)h Nn=0 are optimal for problem

(Ph,∆t), then there exist p(n)h , q

(n)h Nn=0 satisfying the discrete adjoint equations such

that for n = 1, . . . , N − 2:

α(n)h = max

0,min

v(n)h

δ2(2q(n)

h − q(n+1)h ), α1

,

µ(n)h = max

0,min

u(n)h

δ1(2p(n)

h − p(n+1)h ), µ1

,

and for n = N − 1:

α(N−1)h = max

0,min

2v(N−1)h q

(N−1)h

δ2α1,

µ(N−1)h = max

0,min

2u(N−1)h p

(N−1)h

δ1, µ1,

and for n = N : α(N)h = µ

(N)h = 0.

Proof. The proof is standard and is similar to the proof of the correspondingcontinuous result.

6.2. Construction of targets. For the specific target functions (u, v) in ournumerical simulations we chose the skin patterns of the Emperor Angelfish (Pomacan-thus imperator), which is widespread in central and western Pacific. This angelfishwas chosen because of the complex series of stripes and spots on the skin. Our start-ing point prior to pre-processing of the image was a high resolution JPEG image(1050× 750 (3.5in by 2.5in at 300ppi), Copyright Robert Fenner, WetWebMedia.com).

The pre-processing steps are illustrated in Figure 6.1. The original image is shownin Figure 6.1(a). The image is then cropped, which excludes the portion of the tailwith no pattern and the background details (Figure 6.1(b)). The cropped image isalso fitted into a square [0, 2] × [0, 2], which defines the domain Ω. The image inFigure 6.1(b) is then converted to a grayscale image with 256 shades of gray (Figure6.1(c)). This image is then interpolated (2D bicubic) onto the finite element mesh,with a large number of equilateral triangles (Figure 6.1(d)).

As we have no knowledge of the actual maximum and minimum values of theangelfish image, we set the target functions u and v equal to the image scaled between±10% of the equilibrium solutions (i.e., solutions corresponding to f = g = 0) of uand v respectively. The equilibrium solutions (us, vs) are calculated via

us = α(0) + r

µ(0) , vs = ru2s

α(0) ,


where α(0) and µ(0) are the initial guesses for α and µ in the discrete optimal con-trol procedure (see Section 6.3). Finally, we also reverse the grayscale of the targetfunctions for better visibility of the fine structure of the pattern.

(a) (b)

(c) (d)

Fig. 6.1: Pre-processing of a JPEG image of the Emperor Angelfish (P. imperator): (a) Originalimage, (b) Image cropped and fitted into a square, (c) Conversion to gray scale, (d) Image interpolatedonto a Finite Element mesh.

6.3. Discrete optimal control procedure. With the above numerical meth-ods we force the solution of the RD system to match the stationary target function atthe final time T . This necessitates appropriate choices of the weights in the discretecost functional (6.2). By making the weights γi large in relation to the weights βi weplace more emphasis on the solutions matching the target functions at the final time.This makes sense in the context of embryogenesis as we are more interested in thepattern formation at the end of the developmental period than at earlier stages.

To approximate the inverse problem we apply a variable step gradient algorithm[25, 29, 12] yielding a sequence of discrete approximations of

(µ(k), α(k)), (u(k), v(k))k∈N

to the optimal parameters (µ∗, α∗) and corresponding solutions (u∗, v∗). The sensitiv-ities of the system (2.1a)-(2.1b) and cost functional (3.1) with respect to the parame-ters (µ, α) are used to compute the Lagrange multipliers, satisfying the adjoint systemthat marches backward in time. The Lagrange multipliers are then used in a variablestep gradient algorithm (Algorithm 1) to minimize the cost functional. The imple-mentation is straightforward, although computationally intensive. The bulk of the


computational costs are found in the backward-in-time solution of the adjoint systemand the forward-in-time solution of the state system. We begin by making an initialguess for the parameters (µ(0), α(0)) and the step length λ. Then for each iterationk of the gradient method we solve the nonlinear RD system for u(k), v(k), and storethe cost J (µ(k), α(k)). We also compute the adjoint variables (p(k), q(k)), determinedJ (µ(k),α(k))d(µ(k),α(k)) , the total derivative of J with respect to the vector (u(k), v(k)), and

take a step along this direction using the appropriate step length, provided the costfunctional decreases. If the cost functional fails to decrease, then the step is rejectedand the step length decreased. If the step length is accepted, then the parameters(µ(k + 1), α(k + 1)) are updated using a standard gradient update

(µ(k + 1), α(k + 1)) = (µ(k), α(k))− λ(k)dJ (µ(k), α(k))d(µ(k), α(k)) . (6.3)

In the following algorithm we omit the subscript ‘h′ for notational convenience in thediscrete variables: unh, vnh , pnh, qnh , µnh, αnh.

Algorithm 1 Variable Step Gradient Algorithminitialization

k ← 0;RelError ← 10;choose initial guess (αn(0), µn(0)) for n = 1, . . . , N ;choose λ = 1, and tol;solve (6.1) for (un(0), vn(0)) with (u0, v0) for n = 1, . . . , N ;evaluate JN (0) using (6.2);λ← 2λ/3;

end initializationmain loop

while RelError > tol doλ← 3λ/2;k ← k + 1;solve (10.1-10.4) for (pn(k), qn(k)) with (pN (k), qN (k)) for n = N −1, . . . , 0;update (αn(k), µn(k)) using (6.3) for n = 1, . . . , N ;solve (6.1) for (un(k), vn(k)) with (u0, v0) for n = 1, . . . , N ;evaluate JN (k) using (6.2);while JN (k) ≥ JN (k − 1) do

λ← λ/2;update (αn(k), µn(k)) using (6.3) for n = 1, . . . , N ;solve (6.1) for (un(k), vn(k)) with (u0, v0) for n = 1, . . . , N ;evaluate JN (k) using (6.2);

end whileRelError ←

∣∣JN (k)− JN (k − 1)∣∣ / ∣∣JN (k)

∣∣end while

end main loop


7. Numerical experiments. To illustrate the success of our image-driven,PDE-constrained optimization procedure, we present numerical results in two spacedimensions. We used a nonuniform triangulation of the angelfish domain Ω with17,904 nodes and 35,280 triangles, and numerically solved the optimal control prob-lem up-to time T = 10 with uniform time steps ∆t = 1×10−8 (1-SBDF) and ∆t = 0.01(2-SBDF). In all experiments we chose β1 = β2 = 0 to place more emphasis on solu-tions matching the target at the final time T (see comments in Section 6.3).

First experiment:To obtain a good match between u and u, we chose γ2 = 1, γ1 = 0 in the discrete

cost functional (6.2) (see comments in Section 6.3). Figure 7.1 shows a snapshot atT = 10 of the optimal solution u, the target function u, and the optimal parametersα and µ (see the caption for parameter values). The controlled solution u clearlymatches the target u very well. In order to verify the convergence of the discreteoptimal control procedure we also plotted the (discrete) cost functional with iterationcount (Figure 7.2). The cost approaches zero, indicating a good match between theoptimally controlled solution and the corresponding target.

(a) (b)

(c) (d)

Fig. 7.1: (a) Controlled solution u, (b) stationary target function u, (c) optimal parameter α, and(d) optimal parameter µ, at time T = 10. Parameter values: Du = Dv = 0.01, r = 10, β1 = 0,β2 = 0, γ1 = 1, γ2 = 0, δ1 = 1× 10−6, δ2 = 1× 10−6. Initial data: µ = 3, α = 30, u0 and v0 smallrandom perturbations (±10%) of the steady state solutions. Time steps: 1 × 10−8 (1-SBDF), 0.01(2-SBDF). Mesh: 17904 nodes and 35280 triangles.


Fig. 7.2: Change in cost functional with iteration count

Second experiment:We were unable to obtain a good match between v and v with the choice γ2 =

1, γ1 = 0, because the discrete optimal control algorithm did not converge, regardlessof the starting values of the parameters chosen.

We comment that as we effectively have only a single target function (but scaleddifferently for u and v - see Section 6.2), it makes no sense to try and simultaneouslymatch u to u and v to v with the choice γ1, γ2 both non-zero (additional experimentsrevealed that the discrete optimal control algorithm did not converge in these cases).We were also interested in knowing how the parameters change over the time interval[0, T ]. The results of animations demonstrated that both key parameters remainedapproximately constant throughout this time interval except for a rapid change in µnear t = T (see Figures 7.1(c) and 7.1(d)).

8. Conclusions. In this article we considered parameter estimation for a well-known reaction-diffusion (RD) system for patterning with a PDE-constrained image-driven optimization procedure using optimal control theory. Unlike previous studies,the parameters are allowed to vary in both time and space. With this added freedomit is possible to widen the class of models that give rise to pattern formation, and moveaway from the Turing model with its vastly restricted parameter space for patterning[45]. This is demonstrated by our numerical results (Figure 7.1) using equal diffusioncoefficients, as a necessary condition for the generation of diffusion-induced instability(‘Turing patterns’) is unequal diffusion coefficients [63]. The novelty of our work isalso due to the manner in which we constructed a practical target function for theoptimal control algorithm, corresponding to the actual image of a marine angelfish.For concreteness we focussed on the classic Gierer-Meinhardt RD system for patternformation, with two key distributed parameters α and µ.

The mathematical formulation, analysis, and numerical solution of the optimalcontrol problem was presented. After undertaking the mathematical analysis of theoptimal control problem, numerical solutions were obtained with the aid of a ‘lumpedmass’, Galerkin finite element method with piecewise linear continuous basis func-tions. The time-stepping procedure was based on a 2nd order, 3-level, implicit-explicitscheme, which is particularly well suited to the effective approximation of RD systemsfor pattern formation. The numerical results in Figure 7.1 illustrate the success of


a variable step gradient algorithm to identify the space-time distributed parametersneeded to drive the solution of the Gierer-Meinhardt system close to the skin patternof a marine angelfish.

The numerical results suggest that pattern selection in the Gierer-Meinhardt sys-tem depends more on changes in µ than on changes in α. Firstly, we were unable tomatch v and v with the choice γ1 = 0 and γ2 = 1 in the discrete cost functional (6.2).The parameter α occurs in the 2nd equation (2.1b) for v, and with a weak dependanceof pattern selection on α we have effectively only one control in the system (i.e., µin equation (2.1a)). It would appear that the coupling between equations (2.1a) and(2.1b) is not sufficiently strong for us to match v and v via control of µ in the firstequation. The question of whether one can control the whole system with only onecontrol is not a simple one and requires additional investigations. Secondly, whenmatching u and u with the choice γ1 = 0 and γ2 = 1, the parameter α remainedconstant throughout the space-time cylinder Q. These results are consistent withthe results obtained in [23], where parameter identification for the Gierer-Meinhardtsystem for the constant parameter case was investigated. In [23] it was found that theparameter µ is the key parameter in determining pattern selection, while the patternis relatively insensitive to changes in the parameter α.

It is important to note that the Gierer-Meinhardt model is not based on realkinetics and is used in our paper for the purpose of illustrating the methodology forparameter identification in nonlinear RD equations; thus no biological implicationsof our results should be made. However, we can conclude the following: by allowingthe parameters to depend on space and time, the optimal control procedure forcedthe solution u of the Gierer-Meinhardt RD system to match an arbitrary skin patternfrom nature in a virtually point-wise sense.

Acknowledgments. The authors would like to thank Herb Kunze and AlanWillms (University of Guelph, Canada), for their helpful comments.

9. Appendix A: Regularized system. In order to construct stable finite ele-ment approximations (details omitted), we introduce the following regularized versionof (2.1a)-(2.1b):

∂uε∂t = Du∆uε + ru2

ε

vε+εu2ε+ε− µ(x, t)uε + r,

∂vε∂t = Dv∆vε + ru2

ε

1+εu2ε− α(x, t)vε,

with ε = 10−6. We confirmed that there is little difference in solutions with ε = 0. Theglobal existence and uniqueness of the classical solutions of this regularized systemfollows in a straightforward manner from the theoretical framework of Morgan [44].

10. Appendix B: Time stepping scheme for the adjoint equations. Weomit the subscript ‘h’ in the discrete adjoint variables pnh, qnh for notational conve-


nience. The adjoint variables pn, qn satisfy for the first time step:

2p0 − 4p1 + p2

2∆t −Du∆p0

=(r

2u1(|v1|+ ε)(|v1|+ ε|u1|2 + ε)2 − µ

1)· (2p1 − p2) + 2r u1

1 + ε|u1|2· (2q1 − q2)

+β1(u1 − u1),

2q0 − 4q1 + q2

2∆t −Dv∆q0

= r|u1|2sign(v1)

(|v1|+ ε|u1|2 + ε)2 · (−2p1 + p2) + α1(−2q1 + q2) + β2(v1 − v1),

(10.1)and for n = 2, . . . , N − 2 we have:

3pn−1 − 4pn + pn+1

2∆t −Du∆pn−1

=(r

2un(|vn|+ ε)(|vn|+ε|un|2+ε)2−µ

n

)·(2pn − pn+1)+2r un

1+ε|un|2 ·(2qn−qn+1)

+β1(un − un),

3qn−1 − 4qn + qn+1

2∆t −Dv∆qn−1

= r|un|2sign(vn)

(|vn|+ε|un|2+ε)2 ·(−2pn+pn+1)+ αn(−2qn+qn+1) + β2(vn−vn),

(10.2)and for n = N − 1:

3pN−2 − 4pN−1

2∆t −Du∆pN−2

=(r

2uN−1(|vN−1|+ ε)(|vN−1|+ ε|uN−1|2 + ε)2 − µ

N−1)· (2pN−1 − 2pN )

+2r uN−1

1 + ε|uN−1|2· (2qN−1 − qN ) + β1(uN−1 − uN−1),

3qN−2 − 4qN−1

2∆t −Dv∆qN−2

= r|uN−1|2sign(vN−1)

(|vN−1|+ ε|uN−1|2 + ε)2 · (−2pN−1 + pN )

+αN−1(−2qN−1 + qN ) + β2(vN−1−vN−1),

(10.3)

and for n = N :3pN−1 − 2pN

2∆t −Du∆pN−1 = β1(uN − uN ),

3qN−1 − 2qN

2∆t −Dv∆qN−1 = β2(vN − vN ),

supplemented with the final conditions:

pN = γ1(uN − uN ), qN = γ2(vN − vN ). (10.4)


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